ANYONE WHO HAS FLOWN over a waterway at a low or moderate altitude will have noticed the wake patterns left by moving boats. Each pattern is intricate, with a variety of waves contained within a V shape that travels along with the boat as if it were being towed. The general appearance of the wake may suggest that it is a shock wave similar to those shed by supersonic aircraft. (Although a shock wave in air is normally invisible, it can be photographed with special equipment when an aircraft model is mounted in a wind tunnel.) The shock wave also forms a V whose apex seems to be attached to the craft generating it.

In spite of the general resemblance of the two types of pattern, they are produced by different means. One clue to this fact lies in the angle between the arms of the V's. In a shock wave the angle depends on the relative speeds of the aircraft and of sound at the aircraft's altitude. If the aircraft's speed increases, the angle decreases, narrowing the V. Decreasing the speed widens the V, until it disappears when the plane's speed drops below the speed of sound.

If a wake pattern were only a shock wave, the angle of its V would also depend on speed. Actually the angle is always about 39 degrees, regardless of the boat's speed. As a matter of fact, anything at all, from ducks to giant petroleum tankers, creates a wake pattern with that same angle when it moves through water. Wakes also differ from shock waves in that they have a complex structure. The arms of the V are not single, long waves; instead they are made up of a series of short lengths of waves, which accounts for their feathery appearance [see Figure 2]. Extending between the arms there are curved transverse waves that seem to travel along with the boat.

The mechanism producing a boat's wake was first worked out by Lord Kelvin in 1887. The solution was quite challenging, requiring the invention of a mathematical tool called the method of stationary phase. The method enabled Kelvin to approximate how the waves created by a boat interfere with one another. Research into wake production has continued to the present day, spurred by the fact that much of a ship's energy consumption goes to, generating the wake pattern.

If you visualize a water wave, chances are that you imagine a sinusoidal wave traveling over the water surface at a certain speed. The wave, which is often called a phase wave, is characterized by crests and valleys . that are evenly spaced and of uniform height and depth. The horizontal distance between adjacent crests is the wavelength. The frequency at which successive crests pass a given checkpoint is the wave's frequency, which is inversely proportional to the wavelength. The wave's amplitude is the height of a crest above a calm surface.

Although a phase wave is a convenient model, it is actually never seen on water because of a subtle property of water waves: their speed depends on their wavelength or (equivalently) their frequency. This fact guarantees that whenever you disturb a water surface, you see a complicated pattern, rather than a sinusoidal wave, spreading over the water. The pattern is the result of the interference of many phase waves, each traveling at a different speed; although individual phase waves are present, they are hidden from view.

You can produce a water wave that is approximately a phase wave by arranging for a machine to produce periodic oscillations in the water at a certain frequency. (You must also eliminate the possibility that reflections will send waves back into the region you are examining.) If you then look at the resulting wave at some point that is not too near the machine, it is approximately sinusoidal. Its frequency matches that of the machine.

Suppose you measure the speed of the wave and then adjust the machine to generate a wave that has a higher frequency and therefore a shorter wavelength. You will find that the new wave travels slower than the preceding one. With enough experimentation you will discover that the speed of a phase wave depends on the square root of its wavelength. When the speed of a wave depends on the wavelength in any way, the wave is said to be dispersive. It is the dispersive property of water waves that shapes the wake left by a moving boat.

Textbooks often introduce dispersion with an example in which two phase waves differ only slightly in wavelength and wave speed. If the waves were sent over a water surface, they would interfere, giving rise to a moving pattern within which the phase waves are hidden. The wavy structure of the water surface is gathered into groups that move in a parade in the direction of the phase waves [see illustration below]. Crests and valleys progress through each group with twice the group's speed, appearing first at the rear of the group, growing larger toward the center of the group and then dying out when they reach the front.

The crests and valleys mark where the phase waves happen to be approximately in phase, or in step, so that they add constructively to form high crests and deep valleys. Toward each side of the group they are partially out of phase and interfere destructively, creating crests and valleys that are less pronounced. Between the groups the destructive interference is complete. Notice that although the groups seem to move as a whole (particularly if you are too far away to make out their internal structure), they are actually being recreated continuously as fresh sections of the phase waves progress through them. What you see move and what seem to persist are the regions of constructive interference.

If the two phase waves are replaced by phase waves that have a narrow range of wavelengths, the interference pattern is similar, with crests and valleys passing through the groups at twice the group speed. The pattern soon differs from the preceding example, however, because the phase waves with longer wavelengths outrun those with shorter wavelengths owing to dispersion. As the groups move away from the source of the waves, they change shape continuously. One way to describe the groups is as regions where the component phase waves are constantly in phase-hence Kelvin's 9 term "constant phase."

Most disturbances of a water surface are more complicated than the i preceding examples because they create phase waves that have a wide range of wavelengths. Suppose a stone falls into a large pond while you examine the water surface reasonably far from the fall. For a short time, perhaps tens of seconds, the disturbance sends out phase waves continuously. The ones with long wavelengths (and consequently with high speeds) are the first to begin passing you. With time, phase waves that move more and more slowly begin to pass you. Although there is soon a host of passing waves, the water surface remains calm because the waves all interfere destructively: every crest of a given phase wave is canceled by the valleys of other phase waves.

The earliest evidence of activity reaches you when the groups begin to pass. Each group travels at half the speed of its associated phase waves. The first group, the fastest, is produced by the phase waves having the longest wavelengths and therefore the highest speeds. With time, groups that are traveling ever more slowly go by, associated with phase waves that have progressively shorter wavelengths and lower speeds. Whereas in the first example there was a parade of identical groups, the stone gives rise to a succession of different groups that blend into one another.

The transition between groups is so smooth that you may have trouble seeing it, particularly since the action passes you quickly. If you photograph the display to freeze the action and measure the crest-to-crest separations, you will see that the faster groups are ahead of the slower ones. The separation is greatest in the region farthest from the impact of the stone, indicating that the phase waves with the longer wavelengths are responsible for that region. The separation gets progressively shorter as you examine the water surface nearer the impact point, indicating that phase waves with shorter wavelengths come into play there. By making a series of photographs you can see the groups stretch out as the faster phase waves race away from the slower ones, until energy losses and the spread of the waves finally eliminate all action. A stone produces a single point disturbance. A moving boat creates a continuous series of point disturbances, each disturbance sending out expanding groups that interfere with one another to form the characteristic wake. Because Kelvin's solution to the interference problem is complex, I shall first give a much simpler account that was presented in 1984 by Frank S. Crawford, Jr., of the University of California at Berkeley.

Crawford starts out by considering the shock wave a traveling boat would produce if water waves were not dispersive. In the illustration below a boat has traveled from A to B at a constant speed. Its disturbance at A sent out circular phase waves (all moving at the same speed, since we are ignoring dispersion). Phase waves were also created by the boat at each point between A and B, producing a composite pattern of circles of varying size, the boundaries of which form a V-shaped shock wave. By convention the angular extent of a shock wave is measured by the half-angle of the V. In the illustration the sine of that angle is equal to the ratio of the speed of the waves to the speed of the boat.

The fictional shock wave serves as a benchmark for constructing the actual wake created by dispersion. Although the boat's disturbance creates phase waves with a wide range of wavelengths, apparently the only waves that are important are those with speeds no greater than the boat's speed. First consider a small packet of phase waves that differ only slightly in wavelength and that travel at or very close to, say, .866 of the boat's speed. When the boat reaches B, the phase waves that were generated at A reach Won the arm of a fictional shock wave that, according to the equation, is at an angle of 60 degrees to the boat's path. Identical packets that were produced by disturbances between A and B (say at point C) also reach the arm of the fictional shock wave just then.

The shock wave is said to be fictional because the water along the arm is completely calm owing to the destructive interference of the phase waves there. The activity the phase waves do create is back in the groups that have traveled outward at half the speed of the phase waves [below left], and so the group associated with the phase waves reaching W is halfway between A and W. Draw a line between B and that halfway point. The line, which can be called the wake line, marks the current location of groups associated with the chosen phase waves that were generated at points between A and B.

If the boat had generated only these chosen phase waves, the wake would be contained by the wake line, and the wake angle, measured between the line and the boat's path, would be 19.1 degrees. The calculation of the wake angle goes as follows. Let L be the AB distance. Since the phase waves were chosen to travel at .866 times the boat speed, W must be .866 L from A, and the midpoint between A and W must be .433 L from A. Since the right triangle formed by A, W and B has one angle of 60 degrees, the angle WAB must be 30 degrees. The wake angle can then be computed by applying trigonometric rules to the triangle created by A, B and the midpoint of AW.

If you calculate the wake angles associated with other speeds chosen for the phase waves, you will find that the wake angle has a maximum value of 19.5 degrees, which corresponds to phase waves that travel at .82 times the boat's speed. The wake line at this maximum angle marks the boundary of the pattern. It is most apparent in the pattern because a wide range of phase waves contribute groups on or just inside the line, generating prominent crests and valleys there. The full angle of the V, twice the maximum wake angle, is 39 degrees, which matches what is observed in actual wake patterns.

Crawford went on to consider the transverse waves in the wake pattern. They are due to the spreading out of groups toward the boat in circular arcs. Since the pattern remains stationary with respect to the boat, it must be dominated by a packet of phase waves that travel at, or only slightly above or below, the speed of the boat. The associated group, of course, travels at half the boat's speed. The packet created at A, then, creates a group that is halfway along AB when the boat reaches B. The other groups forming the transverse waves have a similar relation to the boat. If a group currently lies at a distance x behind the boat, it must have been created by a disturbance point that is now 2x behind the boat. Furthermore, the transverse wave that group forms must now have a radius of x. The composite pattern seems to be dragged along by the boat but is actually being re-created continuously by fresh phase waves as the transverse waves expand.

In 1957 James Johnston Stoker of New York University and Harvey Douglas Keith of the University of Bristol independently published simplified versions of Kelvin's analysis. In their versions the groups sent out from A are interfered with by groups sent out from other points. When the boat reaches B, the only surviving groups from A lie in a circle, with A on the circumference of the circle at the rear (with respect to the motion of the boat). The group at the front of the circle, produced by phase waves traveling at the speed of the boat, lies halfway between A and B. Other parts of the circle are dominated by groups associated with slower phase waves.

The circles are not waves; they merely indicate the location of the groups. To determine the orientation of a group at any point on a circle, draw a short line through the point in such a way that it is perpendicular to from the point to the disturbance point associated with the circle. The short line represents the visible waves in the group at that point on the circle.

The illustration below shows several circles, each of which is linked to a disturbance point at the rear of the circle. Straight lines drawn along the sides of the circles define the outer limits of the wake pattern. One circle is chosen for a calculation of the angle of the lines-the wake angle. The disturbance point A at a distance L from the boat is on a circle that has a radius of L/4, the center of which is 3L/4 from the boat. Note that a radial line and the line from the center to the boat form part of a right triangle.

The sine of the wake angle, which lies within the triangle, must then be equal to the ratio of the lines (1/3), and the angle itself must be 19 degrees 28 minutes.

I have considered the wake patterns for boats traveling in a straight line in deep water. How do the patterns change when the water is shallow or when its depth varies? What happens if the boat is traveling in a curved path? Why does the relative visibility of the transverse waves and the featherlike waves vary from one wake pattern to the next? I leave these matters for you to investigate the next time you fly over a waterway.

Bibliography

WATER WAVES: THE MATHEMATICAL THEORY WITH APPLICATIONS. J. J. Stoker. Interscience Publishers, 1957.

ELEMENTARY DERIVATION OF THE WAKE PATTERN OF A BOAT. Frank S. Crawford in American Journal of Physics, Vol. 52, No. 9, pages 782-785; September, 1984.

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